Topology General Exam
August 16, 2010

Name:

Instructions: This is a four hour exam and 'closed book'. There are eight problems.

(a) Let $T \subset \mathbb{R}^{5}$ be a closed subspace homeomorphic to $\mathbb{R}^{2}$ . Explain why $T$ will be a retract of $\mathbb{R}^{5}$ .
(b) View $S^{n}$ as $\mathbb{R}^{n} \cup\{\infty\}$ , so that the open subsets of $S^{n}$ containing $\infty$ are precisely the complements of compact subsets of $\mathbb{R}^{n}$ . Recall that a continuous function $f: \mathbb{R}^{m} \rightarrow \mathbb{R}^{n}$ is called proper if $f^{-1}(C)$ is compact in $\mathbb{R}^{m}$ whenever $C$ is compact in $\mathbb{R}^{n}$ . Show that such a proper map extends uniquely to a continuous function $\bar{f}: S^{m} \rightarrow S^{n}$ .
(c) With $T$ as in part (a), check that the inclusion $i: T \hookrightarrow \mathbb{R}^{5}$ is proper. By contrast, show that no retraction $r: \mathbb{R}^{5} \rightarrow T$ can be proper. (Hint: start by using part (b).)

Let $\mathbb{R}^{\infty}$ denote the union $\mathbb{R} \hookrightarrow \mathbb{R}^{2} \hookrightarrow \mathbb{R}^{3} \hookrightarrow \ldots$ , with the union topology, i.e. $U \subset \mathbb{R}^{\infty}$ is open iff $U \cap \mathbb{R}^{n}$ is open in $\mathbb{R}^{n}$ for all $n$ . Let $\mathbb{R}^{\omega}$ denote the product of a countable number of copies of $\mathbb{R}$ , with the product topology. Check that the evident set theoretic inclusion $i: \mathbb{R}^{\infty} \rightarrow \mathbb{R}^{\omega}$ is continuous, but is not a homeomorphism onto its image.

(a) Describe a connected double cover of $\mathbb{R} P^{2} \vee \mathbb{R} P^{2}$ . (There is more than one correct answer.)
(b) What are the homology groups of your double cover?
(c) What is the fundamental group of your double cover?

Let $M$ be the compact surface with boundary circle $C$ as pictured:
(a) Explain why $M$ is homotopy equivalent to a figure eight. (Hint: $M$ is the torus with a disk removed, and the torus is often represented as a square with opposite edges identified.)
(b) Explain why the inclusion $i: C \hookrightarrow M$ induces the zero homomorphism from $H_{1}(C)$ to $H_{1}(M)$ .
(c) By contrast, explain why $i$ is not null homotopic.

Suppose given a commutative diagram of abelian groups

with exact rows and columns. Show that there are isomorphisms $\operatorname{ker} \alpha \simeq \operatorname{ker} \beta \quad$ and $\quad \operatorname{coker} \alpha \simeq \operatorname{coker} \beta$ .

Recall that an $n$ -dimensional manifold is a Hausdorff topological space $M$ that can be covered by open sets homeomorphic to open sets in $\mathbb{R}^{n}$ . Prove that a compact $n$ -dimensional manifold can be embedded in (i.e. is homeomorphic to a subset of) $\mathbb{R}^{N}$ for large enough $N$ . (Hint: use a partition of unity associated to a finite open cover $U_{1}, \ldots, U_{k}$ of $M$ equipped with embeddings $f_{i}: U_{i} \rightarrow \mathbb{R}^{n}$ .)

Let $C \subset \mathbb{R}^{3}$ be the union of the $x$ -axis and the $y$ -axis. Compute $H_{*}\left(\mathbb{R}^{3}-C\right)$ . (Hint: note that $\mathbb{R}^{3}-C=\left(\mathbb{R}^{3}-x\right.$ -axis $) \cap\left(\mathbb{R}^{3}-y\right.$ -axis $)$ .)

Let $X$ be a Hausdorff space, and $f: X \rightarrow X$ a continuous function such that

$f(x) \neq x$ for all $x \in X$ , and
$f \circ f$ is the identity.
(a) Show that every $x \in X$ has an open neighborhood $W_{x}$ satisfying $f\left(W_{x}\right) \cap W_{x}=\emptyset$ .
(b) Let $\bar{X}=X /(x \sim f(x))$ , with the quotient space topology. Show that the quotient map $q: X \rightarrow \bar{X}$ is a covering map.